... specifically, I'm using getSystemClipboard()... and then doing things like getting and setting content.
At the start of the API doc (Java 7) it says:
Many GUI events may be delivered to user asynchronously, if the opposite is not specified explicitly. As well as many GUI operations may be performed asynchronously. This fact means that if the state of a component is set, and then the state immediately queried, the returned value may not yet reflect the requested change.
... I'm not at all clear how this relates to whether or not Toolkit's methods (and clipboard's) should be run in the EDT...!